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A Mean Field Model for the Quadrupolar Phases of UPd$_3$

UPd$_3$ is known to exhibit four antiferroquadrupolar ordered phases at low temperatures. We report measurements of the magnetisation and magnetostriction of single crystal UPd$_3$, along the principal symmetry directions, in fields up to 33 T. These results have been combined with recent inelastic neutron and x-ray resonant scattering measurements to construct a mean field model of UPd$_3$ including up to fourth nearest neighbour interactions. In particular we find that anisotropic quadrupolar interactions must be included in order to explain the low temperature structures derived from the scattering data.